An Introduction to Elliptic Curves

Elliptic Curves and the Abelian Grou

An Introduction to Elliptic Curves
Elliptic Curves are curves of the form
y2=x3+ax+by^2=x^3+ax+by2=x3+ax+b
This is called a Weierstrass equation. The rational points on this elliptic curve miraculously create an abelian group under an operation called point addition.
Point Addition
Given two points PPP
 and QQQ
, we can calculate a third point P⊕QP \oplus QP⊕Q
 on the elliptic curve itself using a simple algorithm:
The images here are from Joseph Silverman's presentation slides from 2006, which are an excellent reference and can be found here.
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First, we draw the line LLL
 through the points PPP
 and QQQ
, which intersects the curve at a third point RRR
. We then draw a vertical line passing through RRR
 which intersects the curve at a second point we call R′R^\primeR′
. R′R^\primeR′
 is the result of P⊕QP \oplus QP⊕Q
, the point addition of PPP
 and QQQ
.
Adding a Point to Itself
We can also add a point PPP
 to itself, but how can we do that with infinitely many lines passing through? We simply take the tangent to the curve and find the point RRR
 it intersects at before mirroring the point to R′=2PR^\prime = 2PR′=2P
.
The Point at Infinity
Rarely, you may find yourself adding PPP
 to −P-P−P
, the point directly below PPP
. In this case there is no third point of intersection and we say the result of point addition is O\mathcal{O}O
, the point "at infinity".
Addition Rules

Baby Step, Giant Step

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